We are given that $\dfrac{dy}{dx}=e^{5y}$. Find an expression for $\dfrac{d^2y}{dx^2}$ in terms of $x$ and $y$. $\dfrac{d^2y}{dx^2}=$
Solution: Notice that the equation defines $y$ implicitly—we don't have an explicit expression for $y$ in terms of $x$. So we will have to use implicit differentiation. If we differentiate the equation once, we will have $\dfrac{d^2y}{dx^2}$ on the left-hand side of the equation. This is what we get after we differentiate each side of the equation: $\dfrac{d^2y}{dx^2}=5e^{5y}\cdot\dfrac{dy}{dx}$ The expression we got for $\dfrac{d^2y}{dx^2}$ isn't in terms of only $x$ and $y$ because it includes $\dfrac{dy}{dx}$ in it. Let's plug the original equation ${\dfrac{dy}{dx}=e^{5y}}$ in the new equation to obtain an expression in terms of $y$ alone: $\begin{aligned} \dfrac{d^2y}{dx^2}&=5e^{5y}\cdot{\dfrac{dy}{dx}} \\\\ &=5e^{5y}\cdot{e^{5y}} \\\\ &=5e^{10y} \end{aligned}$ In conclusion, this is an expression for $\dfrac{d^2y}{dx^2}$ in terms of $x$ and $y$ : $\dfrac{d^2y}{dx^2}=5e^{10y}$